Citation: Qi, W.; Li, Y. Almost
Anti-Periodic Oscillation Excited by
External Inputs and Synchronization
of Clifford-Valued Recurrent Neural
Networks. Mathematics 2022, 10, 2764.
https://doi.org/10.3390/
math10152764
Academic Editors: Krzysztof
Ejsmont, Aamer Bilal Asghar,
Yong Wang and Rodolfo Haber
Received: 7 July 2022
Accepted: 2 August 2022
Published: 4 August 2022
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Article
Almost Anti-Periodic Oscillation Excited by External Inputs and
Synchronization of Clifford-Valued Recurrent Neural Networks
Weiwei Qi and Yongkun Li *
Department of Mathematics, Yunnan University, Kunming 650091, China
* Correspondence: yklie@ynu.edu.cn
Abstract:
The main purpose of this paper was to study the almost anti-periodic oscillation caused
by external inputs and the global exponential synchronization of Clifford-valued recurrent neural
networks with mixed delays. Since the space consists of almost anti-periodic functions has no
vector space structure, firstly, we prove that the network under consideration possesses a unique
bounded continuous solution by using the contraction fixed point theorem. Then, by using the
inequality technique, it was proved that the unique bounded continuous solution is also an almost
anti-periodic solution. Secondly, taking the neural network that was considered as the driving system,
introducing the corresponding response system and designing the appropriate controller, some
sufficient conditions for the global exponential synchronization of the driving-response system were
obtained by employing the inequality technique. When the system we consider degenerated into a
real-valued system, our results were considered new. Finally, the validity of the results was verified
using a numerical example.
Keywords:
almost anti-periodicity; Clifford-valued recurrent neural network; mixed delay; global
exponential synchronization
MSC: 34K14; 34K24; 92B20
1. Introduction
A recurrent neural network is a recurrent neural network that takes sequence data
as input and recurses in the evolutionary direction of the sequence, with all nodes con-
nected in a chain. Research on recurrent neural networks started in the 1980s and 1990s
and developed into one of the deep learning algorithms in the early 2000s [
1
]. Recurrent
neural network has been applied in natural language processing, such as speech recogni-
tion, language modeling, machine translation and so on. It is also used for various time
series forecasting. These applications of recurrent neural networks are closely related to
their dynamic behavior. Therefore, the dynamics of recurrent neural networks have been
extensively studied over the past few decades.
As a high-dimensional neural network, the Clifford-valued neural network not only
includes real-valued, complex-valued and quaternary-valued neural networks as its special
cases, but also has greater advantages than low-dimensional neural networks in deal-
ing with multidimensional data. In recent years, they have attracted more and more
attention [2–11].
On the one hand, it is well known that periodic and almost periodic oscillations are
the focus of qualitative research on differential equations [
12
–
14
]. Anti-periodic oscillation
is a special form of periodic oscillation, but it can reflect a particularly accurate oscillation
and has many important applications, such as in interpolation problems [
15
,
16
], wavelet
theory [
17
], neural networks [
18
–
27
], etc. In the past decade, anti-periodic oscillation has
been widely studied. Recently, the concept of almost anti-periodic functions was proposed
in [
28
], which is a generalization of the concept of anti-periodic functions. Any anti-
periodic function is an almost anti-periodic function, but an almost periodic function is not
Mathematics 2022, 10, 2764. https://doi.org/10.3390/math10152764 https://www.mdpi.com/journal/mathematics
